Measure-preserving transformations between the same measure space are sometimes called of the measure space.

Remarks:

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The fact that a map T:X1⟶X2 is measure-preserving depends heavily on the sigma-algebras𝔅i and measuresμi involved. If other measures or sigma-algebras are also in consideration, one should make clear to which measure space is T:X1⟶X2 measure-preserving.

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Measure-preserving maps are the morphisms on the category whose objects are measure spaces. This should be clear from the next results and examples.

2 Properties

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The composition of measure-preserving maps is again measure-preserving. Of course, we are supposing that the domains and codomains of the maps are such that the composition is possible.

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Let (X1,𝔅1,μ1) and (X2,𝔅2,μ2) be measure spaces and (X1,𝔅1¯,μ1¯) and (X2,𝔅2¯,μ2¯) their completions. If T:(X1,𝔅1,μ1)⟶(X2,𝔅2,μ2) is measure-preserving, then so is T:(X1,𝔅1¯,μ1¯)⟶(X2,𝔅2¯,μ2¯).

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Let (X1,𝔅1,μ1) and (X2,𝔅2,μ2) be measure spaces and T1:X1⟶X1, T2:X2⟶X2 be measure-preserving maps. Then, the product mapT1×T2:X1×X2⟶X1×X2, defined by

T1×T2⁢(x1,x2):=(T1⁢(x1),T2⁢(x2))

is a measure-preserving transformation of (T1×T2,𝔅1×𝔅1,μ1×μ2).

3 Examples

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The identity map of a measure space (X,𝔅,μ) is always measure-preserving.

Every continuoussurjectivehomomorphism between compact Hausdorff is measure-preserving relatively to the normalized Haar measure (see this entry (http://planetmath.org/ContinuousEpimorphismOfCompactGroupsPreservesHaarMeasure)).